Hurwitz theorem
E637304
Hurwitz theorem is a fundamental result in Diophantine approximation that gives an optimal bound on how well any irrational real number can be approximated by infinitely many rational numbers.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hurwitz theorem canonical | 1 |
| Hurwitz's theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7030798 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hurwitz theorem Context triple: [Diophantine approximation, hasKeyResult, Hurwitz theorem]
-
A.
Hermite–Minkowski theorem
The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
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B.
Hurwitz
Hurwitz is a surname of German and Ashkenazi Jewish origin borne by various notable individuals across fields such as mathematics, music, and law.
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C.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
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D.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
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E.
Hurwitz quaternions
Hurwitz quaternions are a specific lattice of quaternions with integer and half-integer components that form a maximal order in the quaternion algebra and provide a natural algebraic framework for understanding representations of integers as sums of four squares.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hurwitz theorem Target entity description: Hurwitz theorem is a fundamental result in Diophantine approximation that gives an optimal bound on how well any irrational real number can be approximated by infinitely many rational numbers.
-
A.
Hermite–Minkowski theorem
The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
-
B.
Hurwitz
Hurwitz is a surname of German and Ashkenazi Jewish origin borne by various notable individuals across fields such as mathematics, music, and law.
-
C.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
D.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
-
E.
Hurwitz quaternions
Hurwitz quaternions are a specific lattice of quaternions with integer and half-integer components that form a maximal order in the quaternion algebra and provide a natural algebraic framework for understanding representations of integers as sums of four squares.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in Diophantine approximation ⓘ |
| appliesTo | irrational real numbers ⓘ |
| assumptionOnVariable | x is irrational ⓘ |
| boundIs | 1/(sqrt(5) q^2) ⓘ |
| characterizes | quality of approximation of irrationals by rationals ⓘ |
| classification | classical theorem in number theory ⓘ |
| codomain | rational approximations ⓘ |
| concerns | approximation of real numbers by rational numbers ⓘ |
| conclusion | there exist infinitely many integers p and q with q > 0 satisfying |x - p/q| < 1/(sqrt(5) q^2) ⓘ |
| domain | real numbers ⓘ |
| extremalCase | conjugate of the golden ratio ⓘ |
| extremalCase | golden ratio ⓘ |
| field |
Diophantine approximation
NERFINISHED
ⓘ
number theory ⓘ |
| gives | optimal bound on rational approximation of irrationals ⓘ |
| hasGeneralization |
Khinchin-type theorems in Diophantine approximation
NERFINISHED
ⓘ
results on Markov numbers ⓘ |
| hasOptimalConstant | 1/sqrt(5) ⓘ |
| hasProofMethod |
continued fraction expansion of irrationals
ⓘ
properties of convergents ⓘ |
| implies |
every irrational has infinitely many very good rational approximations
ⓘ
no irrational can be approximated better than order 1/q^2 with a larger uniform constant ⓘ |
| involvesConstant | sqrt(5) ⓘ |
| isPartOf | classical theory of Diophantine approximation ⓘ |
| namedAfter | Adolf Hurwitz NERFINISHED ⓘ |
| originalLanguage | German ⓘ |
| quantifier |
for every irrational real number
ⓘ
infinitely many rational numbers ⓘ |
| refines | Dirichlet approximation theorem NERFINISHED ⓘ |
| relatedProblem | finding best possible constants in rational approximation inequalities ⓘ |
| relatedTo |
Lagrange spectrum
NERFINISHED
ⓘ
Markov spectrum NERFINISHED ⓘ best Diophantine approximations ⓘ |
| sharpness |
bound cannot be improved for all irrationals
ⓘ
constant 1/sqrt(5) is best possible ⓘ |
| statesThat | for every irrational real number x there exist infinitely many rationals p/q with |x - p/q| < 1/(sqrt(5) q^2) ⓘ |
| topic |
metric Diophantine approximation
ⓘ
rational approximation exponents ⓘ |
| typeOfBound | uniform bound valid for all irrationals ⓘ |
| usedIn |
Diophantine analysis of quadratic irrationals
ⓘ
study of badly approximable numbers ⓘ theory of continued fractions ⓘ |
| usesConcept |
continued fractions
ⓘ
convergents of continued fractions ⓘ |
| yearProved | 1891 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hurwitz theorem Description of subject: Hurwitz theorem is a fundamental result in Diophantine approximation that gives an optimal bound on how well any irrational real number can be approximated by infinitely many rational numbers.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Montel's theorem
this entity surface form:
Hurwitz's theorem